**4. Entropy Generation**

The buoyance induced convection in a closed chamber discovers significant awareness in thermal engineering applications. However, the practice of entropy generation supports to spot the ideal conditions for many applications. Since the generation of entropy is as a result of the irreversible procedure of transfer of heat and viscosity, generation of entropy can be estimated from the well-known thermal and velocity fields.

The entropy generation is expressed by two quantities, i.e., heat transfer (first term in below equation) and liquid friction (last term in below equation) [18,21,22,34].

$$S\_{\rm Cent} = \frac{k\_{\rm rf}}{T\_c^2} \left[ \left( \frac{\partial \mathcal{Q}}{\partial x} \right)^2 + \left( \frac{\partial \mathcal{Q}}{\partial y} \right)^2 + \left( \frac{16 \alpha^\* \mathcal{C}\_0^3}{3 \mathcal{K}^2} \right) \left( \left( \frac{\partial \mathcal{Q}}{\partial x} \right)^2 + \left( \frac{\partial \mathcal{Q}}{\partial y} \right)^2 \right) \right] + \left( \frac{\mu\_{\rm rf}}{T\_c} \right) \left\{ 2 \left[ \left( \frac{\partial \mathcal{Q}}{\partial x} \right)^2 + \left( \frac{\partial \mathcal{Q}}{\partial y} \right)^2 \right] + \left( \frac{\partial \mathcal{q}}{\partial y} + \frac{\partial \mathcal{q}}{\partial x} \right)^2 \right\} \tag{24}$$

The dimensionless entropy generation is acquired by using (10)

$$S\_{\text{total}} = S\_{\text{heat}}^{\star} + S\_{fluid}^{\star}$$

$$S\_{\text{heat}}^{\star} = \left(\frac{k\_{nf}}{k\_f}\right) \left(1 + \frac{4Rd}{3}\right) \left[\left(\frac{\partial T}{\partial X}\right)^2 + \left(\frac{\partial T}{\partial Y}\right)^2\right] \tag{25}$$

$$S\_{fluid}^{\*} = \phi\_2 \left(\frac{\mu\_{nf}}{\mu\_f}\right) \left\{ 2 \left[ \left(\frac{\partial \mathcal{U}}{\partial X}\right)^2 + \left(\frac{\partial \mathcal{V}}{\partial Y}\right)^2 \right] + \left(\frac{\partial \mathcal{U}}{\partial Y} + \frac{\partial \mathcal{V}}{\partial X}\right)^2 \right\} \tag{26}$$

where φ2 = *U*0 <sup>θ</sup>0*L*<sup>2</sup> . The global entropy generation attains by integrating the local entropy production inside the chamber.

$$SG\_{\text{total}} = \int\_{V} S\_{\text{total}}(X, \mathcal{Y}) dA \tag{27}$$

The local Bejan number states the strength of generation of entropy owing to thermal transference irreversibility. It is derived as

$$B\varepsilon\_{\rm loc} = \frac{S\_{\rm hcat}^{\prime}}{S\_{\rm total}} \tag{28}$$

For any point in the chamber, when *Beloc* > 12 , the heat transfer irreversibility is dominating. When *Beloc* < 12 , the liquid friction irreversibility dominates. If *Beloc* = 12 , the thermal and viscous irreversibilities are equal. The average value of Bejan number demonstrates the relative importance of the thermal energy transfer irreversibility for the entire chamber.

$$Be^\* = \frac{\int\_A Be\_{\text{loc}}(X, Y)dA}{\int\_A dA} \tag{29}$$
